After replacing Neil Lennon last summer, the Australian oversaw mixed results as he tried to put together a new team and implement his philosophy.
But after trailing Rangers by six points at the end of 2021, Postecoglou's team have now added consistency and soared three points clear of their city rivals in the cinch Premiership.
"Very good indeed, excellent," former Hoops manager O'Neill said when asked about the progress made by the Australian. "From a stodgy start, domestically it's been really good.
"Looking in from a distance, it seemed a big rebuild was required. It was important that some of the good players stayed on, like Callum McGregor, and some of the players he has brought in have done wonderfully well.
"Sticking to your beliefs as a manager is important. That's where Ange gains a great deal of respect for me because he stuck with it and believed in it.
"At the start, when results weren't really there, you start to think maybe he needs to go with a different emphasis or a change of direction, but, no, he stuck to his beliefs and it's bearing fruit domestically.
"It can be very difficult to do that when results are not going so well, so well done to Ange."
Celtic tightened their grip on top spot when they eked out a 3-2 win at home to Dundee a few hours after Rangers had drawn at Dundee United.
O'Neill insists his old club will still face a stern test from a Gers side who issued a reminder of their capabilities by winning 4-2 away to Borussia Dortmund in the Europa League last week.
"It was a big win for Celtic on Sunday after Rangers couldn't get the win against Dundee United," said O'Neill.
"There's a long way to go yet but they've got the three-point lead and the better goal difference. I'm quite sure Rangers will not give it up easily.
"Having done so brilliantly against Dortmund, they will still feel as if they've got a real chance."
O'Neill was promoting the return of the Masters football tournament this summer after an 11-year absence.
The newly-launched 360 Sports TV Masters Cup will be staged at Glasgow's Braehead Arena on July 8 and will include former players - aged 35 or over - from Premier League giants Liverpool and Manchester United, as well as Celtic and Rangers.
O'Neill will be managing the Hoops side in a tournament that will be beamed by 360 Sports TV.
"I can't wait for it to come back to Glasgow where I have many happy memories and many friends," he said.
"It's great that the fans will be able to watch such an iconic tournament once again with such fabulous football heritage. It should be a really fantastic evening."